Describe the Image and kernel of $T$

Question

In a problem I was given the linear transformation $T:\mathbb{R}^n \rightarrow \mathbb{R}$ where $T$ is defined as

$$T(x)=\det \begin{pmatrix} | & | & &| \\ v_1 & v_2&\cdots& x\\ | & | & &| \\ \end{pmatrix} $$

The question asks me to describe the image and kernel of $T$ and determine their dimension. However, when I think about this I keep trying to solve the kernel equation of $T(x)=0$. I think this is infinite because so long as $x$ is linearly dependent on the other columns the $\det = 0$

Edit:

I now get that the kernel is the $\operatorname{span}(v_1, v_2, \ldots v_n)$. Do I then need to try to create a basis of that span to get the dimension of the kernel? Also, how would I go about describing the kernel? Just the span/basis of the other vectors?

Answer 1

You should consider two cases:

• If $\{v_1, \ldots, v_{n-1}\}$ is not linearly independent, then the image is $\{0\}$ and the kernel is $\mathbb{R}^n$.

• Now, suppose it is linearly independent, then your kernel is right, the kernel is $\operatorname{span} \{v_1, \ldots, v_{n-1}\}$. The image is $\mathbb{R}$.